Interactive Decision Making for Multiobjective
Fuzzy Random Programming Problems with
Simple Recourse through a Fractile Model
Hitoshi Yano and Rongrong Zhang
Abstract—In this paper, we focus on multiobjective fuzzy random programming problems with simple recourse through a fractile optimization model, in which fuzzy random variables coefficients are involved in equality constraints, and random variables coefficients are involved in the objective functions. In the proposed method, equality constraints with fuzzy random variables are defined on the basis of a possibility measure and a two-stage programming method. To deal with the objective functions involving random variable coefficients, a fractile op-timization model is applied. For a given permissible possibility level and/or permissible probability levels specified by the decision maker, several kinds of Pareto optimality concepts are introduced. Interactive decision making methods are proposed to obtain a satisfactory solution from among a Pareto optimal solution set. The proposed method is applied to a farm planning problem in the Philippines, in which it is assumed that the amount of water resource in dry season is represented as a fuzzy random variable.
Index Terms—multiobjective programming, simple recourse programming, a possibility measure, fuzzy random variables, a satisfactory solution.
I. INTRODUCTION
During the past six decades, various types of stochastic programming approaches have been proposed to deal with mathematical programming problems with random variable coefficients. Such approaches can be classified into two groups, one is two-stage programming methods [2], [4], [7], [18], [20] and the other is chance constraints methods [3], [15], [16]. In two-stage programming problems, the first-stage is to minimize the penalty cost for the violation of the equality constraints under the assumption that the decision variables are fixed, and the second-stage is to minimize the original objective function and the corresponding penalty cost. For chance constraint programming problems, a proba-bility maximization model and a fractile optimization model were proposed. In a probability maximization model, the probability that the objective function is smaller than a cer-tain target value is maximized. A fractile optimization model can be regarded as a complementary to the corresponding probability maximization model, in which a target variable is optimized under the condition that the decision maker specifies the probability level that the objective function is smaller than the target variable.
Two-stage programming methods have been applied to various types of water resource allocation problems with
Hitoshi Yano is with Graduate School of Humanities and Social Sciences, Nagoya City University, Nagoya, 467-8501, Japan e-mail: [email protected] .
Rongrong Zhang is with Graduate School of Humanities and Social Sciences, Nagoya City University, Nagoya, 467-8501, Japan.
random inflow in future [17], [19]. However, if probability density functions of random variables are unknown or the problem is a large scale one with random variables, it may be extremely hard to solve the corresponding two-stage programming problem. From such a point of view, inexact two-stage programming methods have been proposed [6], [12].
As an extension for multiobjective programming problems, Sakawa et al. [14] proposed an interactive fuzzy decision making method for multiobjective stochastic programming problems with simple recourse. However, in the real world decision making situations, it seems to be natural to consider that the uncertainty is expressed by not only fuzziness but also randomness simultaneously. From such a point of view, interactive decision making methods for multiobjective fuzzy random programming problems have been proposed [9], [10], in which chance constraint methods and a possibility measure are applied to deal with fuzzy random variable coefficients [11].
In this paper, we focus on multiobjective fuzzy random programming problems with simple recourse through a frac-tile optimization model, where the coefficients of equality constraints are defined by LR-type fuzzy random variables [9], [10], and the coefficients of the objective functions are defined by random variables. We propose interactive decision making methods to obtain a satisfactory solution from among a Pareto optimal solution set. In section II, we focus on multiobjective programming problems, in which the coefficients of equality constraints are fuzzy random variables. After the decision maker specifies a permissible possibility level for fuzzy random variables, multiobjective fuzzy random simple recourse programming problems are formulated, and the corresponding Pareto optimal solution concept is defined. To obtain a satisfactory solution from among a Pareto optimal solution set, an interactive algorithm (Algorithm 1) is developed [22]. In section III, we further focus on multiobjective programming problems, in which the coefficients of equality constraints are fuzzy random variables while the coefficients of the objective functions are random variables. After the decision maker specifies not only a permissible possibility level for fuzzy random variables but also permissible probability levels for the objective functions, multiobjective fuzzy random simple recourse programming problems through a fractile optimization model are formu-lated, and the corresponding Pareto optimal solution concept is defined. To obtain a satisfactory solution from among a Pareto optimal solution set, an interactive algorithm (Algo-rithm 2) is developed. From a view point that permissible probability levels and the corresponding objective functions
IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_14
conflict with each other, the another interactive algorithm (Algorithm 4) is also developed, in which the decision maker is requested to specify not permissible probability levels but the membership functions of permissible probability levels. In Algorithm 3, permissible probability levels are automat-ically set as appropriate values through the fuzzy decision [1], [13], [24]. In section IV, to show the efficiency of the proposed method, we apply Algorithm 2 to a farm planning problem in the Philippines [23], in which it is assumed that the amount of water resource in dry season is represented as a fuzzy random variable and the profit coefficients of seven crops are represented as random variables.
II. MULTIOBJECTIVEFUZZYRANDOMPROGRAMMING
PROBLEMS WITHSIMPLERECOURSE
In this section, we focus on multiobjective programming problems involving fuzzy random variable coefficients in the right-hand sides of the equality constraints.
[MOP1]
min
x∈X (c1x,· · · ,ckx) (1)
subject to
Ax=ed (2) where cℓ = (cℓ1,· · ·, cℓn), ℓ = 1,· · · , k are n
dimen-sional coefficient row vectors of objective function, x = (x1,· · ·, xn)T ≥ 0 is an n dimensional decision variable
column vector, X is a linear constraint set with respect to x. A is an (m × n) dimensional coefficient matrix,
e
d= (ed1,· · · ,edm)
T
is anmdimensional coefficient column vector whose elements are fuzzy random variables [11] (The symbols ”-” and ”˜” mean randomness and fuzziness respectively).
In order to deal with fuzzy random variables efficiently, Katagiri et al. [9], [10] defined a special type of a fuzzy random variable based on the concept of LR fuzzy numbers [5], which is called an LR-type fuzzy random variable. Under the occurrence of each elementary event ω,edi(ω) is a realization of an LR-type fuzzy random variableedi, which is an LR fuzzy number [5] whose membership function is defined as follows.
µe di(ω)
(s) =
L
(
bi(ω)−s
αi )
, s≤bi(ω)
R
(
s−bi(ω)
βi )
, s > bi(ω) (3)
where the function L(t) def= max{0, l(t)} is a real-valued continuous function from [0,∞) to [0,1], and l(t) is a strictly decreasing continuous function satisfying l(0) = 1. Also, R(t) def= max{0, r(t)} satisfies the same conditions.
αij(>0)andβij(>0) are called left and right spreads [5]. The mean value bi is a random variable, whose probability density function and cumulative distribution function are defined as hi(·) and Hi(·) respectively. It is assumed that random variablesbi, i= 1,· · ·, mare independent with each
other.
Since it is difficult to deal with MOP1 directly, we introduce a permissible possibility levelγ(0< γ≤1)based on a concept of a possibility measure [5] for the equality constraints (2),
Pos(aix=edi(ω))≥γ, i= 1,· · · , m, (4)
whereai= (ai1,· · ·, ain), i= 1,· · · , maren-dimensional
row vectors ofA. From the property of LR fuzzy numbers, thei-th inequality condition (4) can be transformed into the following two inequalities.
bi(ω)−L−1(γ)αi ≤aix≤bi(ω) +R−1(γ)βi (5)
For the above two inequalities (5), we introduce two vectors
y+ = (y1+,· · ·, y+m)T ≥0,y−= (y1−,· · · , y−m)T ≥0,
where(y+i , yi−)represent the shortage and the excess for the interval (5), and the following relations hold [21].
(1) For the case bi(ω)−L−1(γ)αi > a
ix, it holds that yi+=bi(ω)−L−1(γ)αi−a
ix>0, yi−= 0.
(2) For the case bi(ω) +R−1(γ)βi < aix, it holds that yi+= 0, y−i =aix−(bi(ω) +R−1(γ)βi)>0.
(3) For the case bi(ω)−L−1(γ)αi ≤ a
ix ≤ bi(ω) + R−1(γ)βi, it holds thaty+
i = 0, y−i = 0.
Yano [21] has already formulated fuzzy random simple re-course programming problems using(y+,y−). In this paper,
as a extension of [21], we formulated a multiobjective fuzzy random simple recourse programming problem (MOP2) as follows.
[MOP2]
min
x∈X c1x+E
[ min y+,y−
(
q+1y++q−1y−)]
· · · ·
min
x∈X ckx+E
[ min y+,y−
(
q+ky++q−ky−)]
(6)
subject to
aix+yi+≥bi(ω)−L−1(γ)αi, i= 1,· · · , m
aix−yi−≤bi(ω) +R−
1(γ)βi, i= 1,· · ·, m x∈X,y+≥0,y− ≥0
where
q+ℓ = (qℓ+1,· · ·, q+ℓm)≥0, ℓ= 1,· · · , k (7)
q−ℓ = (qℓ−1,· · ·, q−ℓm)≥0, ℓ= 1,· · · , k (8) are m dimensional weighting row vectors for y+ and y−
respectively. For theℓ-th objective function of (6), the second term can be transformed into follows [21].
E
[ min y+,y−
(
q+ℓy++q−ℓy−)]
=
m
∑
i=1
qℓi+
(
E[¯bi]−aix−L−1(γ)αi
)
+
m
∑
i=1
qℓi+{(aix+L−1(γ)αi)Hi(aix+L−1(γ)αi)
−
∫ aix+L−1(γ)αi
−∞
bihi(bi)dbi
}
+
m
∑
i=1
qℓi−{(aix−R−1(γ)βi)Hi(aix−R−1(γ)βi)
−
∫ aix−R−1(γ)βi
−∞
bihi(bi)dbi
}
def
= dℓ(x, γ) (9)
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In the following, we define the objective functions as:
zℓ(x, γ)def= cℓx+dℓ(x, γ), ℓ= 1,· · ·, k. (10)
Then, a multiobjective programming problem (1) can be reduced to a multiobjective fuzzy random simple recourse programming problem, in which a permissible possibility level γ is a parameter specified by the decision maker.
[MOP3(γ)]
min
x∈X (z1(x, γ),· · · , zk(x, γ)) (11)
Now, we can define a Pareto optimal solution concept for (11).
Definition 1
x∗ ∈ X is said to be a γ-Pareto optimal solution to MOP3(γ), if and only if there does not exist anotherx∈X
such that zℓ(x, γ) ≤ zℓ(x∗, γ), ℓ = 1,· · · , k with strict inequality holding for at least oneℓ.
For generating a candidate of a satisfactory solution which is also a γ-Pareto optimal solution, the decision maker is asked to specify a permissible possibility level γ and the reference objective values ˆzℓ, ℓ = 1,· · ·, k [13]. Once γ
and zˆℓ, ℓ = 1,· · ·, k are specified, the corresponding γ
-Pareto optimal solution, which is in a sense close to his/her requirement or better than that if the reference objective values are attainable, is obtained by solving the following minmax problem [13].
[MINMAX1(ˆz,γ)]
min
x∈X,λ∈R1λ (12)
s.t. zℓ(x, γ)−zℓˆ ≤λ, ℓ= 1,· · · , k (13) The relationships between the optimal solution(x∗, λ∗)of MINMAX1(zˆ,γ) andγ-Pareto optimal solutions to MOP3(γ)
can be characterized by the following theorem.
Theorem 1
(1) If x∗ ∈ X, λ∗ ∈ R1 is a unique optimal solution of
MINMAX1(zˆ,γ), thenx∗∈X is aγ-Pareto optimal solution to MOP3(γ).
(2) If x∗∈X is a γ-Pareto optimal solution to MOP3(γ), thenx∗∈X λ∗def= zℓ(x∗, γ)−zℓ, ℓˆ = 1,· · ·, kis an optimal solution of MINMAX1(zˆ,γ) for some reference objective valueszˆ= (ˆz1,· · · ,zkˆ ).
(Proof)
(1) Assume thatx∗∈X is not aγ-Pareto optimal solution to MOP3(γ). Then, there existsx∈X such that zℓ(x, γ)≤ zℓ(x∗, γ), ℓ = 1,· · ·, k with strict inequality holding for at
least oneℓ. This means thatzℓ(x, γ)−zˆℓ≤zℓ(x∗, γ)−zˆℓ≤ λ∗, ℓ= 1,· · ·, k, which contradicts the fact thatx∗∈X is a unique optimal solution to MINMAX1(zˆ,γ).
(2) Assume thatx∗∈X, λ∗∈R1is not an optimal solution
to MINMAX1(zˆ, γ) for any reference objective values zˆ= (ˆz1,· · · ,zkˆ ), which satisfy the equalities λ∗ =zℓ(x∗, γ)− ˆ
zℓ, ℓ = 1,· · · , k. Then, there exists some x ∈ X, λ < λ∗
such thatzℓ(x, γ)−zℓˆ ≤λ, ℓ= 1,· · ·, k. This means that
zℓ(x, γ) < zℓ(x∗, γ), ℓ = 1,· · · , k, which contradicts the fact thatx∗∈X is aγ-Pareto optimal solution to MOP3(γ). Unfortunately, it is not guaranteed that (x∗, λ∗) is a γ -Pareto optimal solution to MOP3(γ), if (x∗, λ∗) is not unique. In order to guaranteeγ-Pareto optimality, we solve aγ-Pareto optimality test problem for(x∗, λ∗).
Theorem 2
Let x∗ ∈ X, λ∗ ∈ R1 be an optimal solution to
MINMAX1(zˆ,γ), in which λ∗ = zℓ(x∗, γ) − ˆzℓ, ℓ =
1,· · ·, k. Corresponding to the optimal solution x∗ ∈ X, solve the followingγ-Pareto optimality test problem.
max x∈X,ϵ=(ϵ1,···,ϵk)≥0
k
∑
ℓ=1
ϵℓ (14)
subject to
zℓ(x, γ)−zℓˆ +ϵℓ≤λ∗, ℓ= 1,· · · , k
Let xˇ ∈X,ˇϵℓ ≥0, ℓ = 1,· · ·, k be an optimal solution to (14). If ∑kℓ=1ˇϵℓ = 0, then x∗ ∈ X is a γ-Pareto optimal solution to (11).
On the other hand, the partial differentiation of
zℓ(x, γ), ℓ = 1,· · ·, k for xs, s = 1,· · · , n and xt, t =
1,· · ·, ncan be calculated as follows.
∂zℓ(x, γ) ∂xs∂xt
=
m
∑
i=1
qℓi+aisaithi(aix+L−1(γ)αi)
+
m
∑
i=1
q−ℓiaisaithi(aix−R−1(γ)βi) (15)
The Hessian matrix forzℓ(x, γ)can be written as:
∇2zℓ(x, γ) =
m
∑
i=1
qℓi+hi(aix+L−1(γ)αi)·Ai
+
m
∑
i=1
qℓi−hi(aix−R−1(γ)βi)·Ai, (16)
where Ai, i = 1,· · · , m are (n×n)-dimensional matrices defined as follows.
Aidef=
a2i1 · · · ai1ain
..
. . .. ...
ainai1 · · · a2in
, i= 1,· · ·, m (17)
Because of the property of the Hessian matrix for
zℓ(x, γ), ℓ= 1,· · ·, k, the following theorem holds.
Theorem 3
MINMAX1(zˆ, γ) is a convex programming problem. (Proof)
From the definition (17), it holds thatAi=aT
i ·ai. Therefore,
the following relation holds for anyn-dimensional column vectory∈R1.
yTAiy = yT ·(aTi ·ai)·y
= (yT·aTi )·(ai·y)
= (ai·y) T·
(ai·y)≥0
This means that matrices Ai, i = 1,· · ·, m are positive
semidefinite. Because of the assumptions that probability density functionshi(·)≥0, i= 1,· · · , m, andq+ℓi≥0, qℓi−≥
0, ℓ = 1,· · · , k, i= 1,· · · , m, the following relation holds for each of the Hessian matrices∇2zℓ(x, γ), ℓ= 1,· · ·, k.
yT∇2zℓ(x, γ)y
=
m
∑
i=1
q+ℓihi(aix+L−1(γ)αi)·yTAiy
+
m
∑
i=1
q−ℓihi(aix−R−1(γ)βi)·yTAiy≥0
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This means that MINMAX1(zˆ, γ) is a convex programming problem.
The relationship between a permissible possibility level
γ and the optimal objective function value zℓ(x∗, γ) of MINMAX1(zˆ, γ) can be characterized by the following theorem.
Theorem 4
For the optimal solution x∗ ∈ X of MINMAX1(zˆ, γ), the following relation holds.
∂zℓ(x∗, γ)
∂γ
= −
m
∑
i=1
qℓi+∂L −1(γ)
∂γ αi
+
m
∑
i=1
qℓi+∂L −1(γ)
∂γ αiHi(aix
∗+L−1(γ)α
i)
−
m
∑
i=1
qℓi−∂R −1(γ)
∂γ βiHi(aix
∗−R−1(γ)βi)
(18)
Now, following the above discussions, we can present an interactive algorithm to derive a satisfactory solution from among aγ-Pareto optimal solution set to (11).
[Algorithm 1]
Step 1: Set a permissible possibility levelγ= 1.
Step 2: The decision maker sets the initial reference objective values zˆℓ forzℓ(x, γ), ℓ= 1,· · ·, k.
Step 3: Solve MINMAX1(zˆ, γ)and obtain the correspond-ing optimal solution (x∗, λ∗). For the optimal solution x∗, aγ-Pareto optimality test problem is solved.
Step 4: If the decision maker is satisfied with the cur-rent value of the γ-Pareto optimal solution zℓ(x∗, γ), ℓ = 1,· · · , k, then stop. Otherwise, the decision maker updates his/her reference objective valueszℓ, ℓˆ = 1,· · ·, k, and/or a permissible possibility level γ, and return to Step 3.
III. MULTIOBJECTIVEFUZZYRANDOMPROGRAMMING
PROBLEMS WITHSIMPLERECOURSETHROUGH A
FRACTILEMODEL
In this section, we further consider multiobjective pro-gramming problems, in which the coefficients of equality constraints are fuzzy random variables while the coefficients of the objective functions are random variables.
[MOP4]
min
x∈X (c1x¯ ,· · · ,c¯kx) (19)
subject to
Ax=ed (20) wherec¯ℓ= (¯cℓ1,· · · ,cℓn¯ ), ℓ= 1,· · ·, kis anndimensional
random variable coefficient row vectors of the objective function ¯cℓx.x= (x1,· · ·, xn)
T
≥0 is ann dimensional decision variable column vector,X is a linear constraint set with respect tox. Ais an(m×n) dimensional coefficient matrix,de= (ed1,· · ·,edm)
T
is an m dimensional coefficient column vector whose element is an LR-type fuzzy random variableedi [9], [10] defined by (3).
In the following, let us assume that the each element ¯cℓj
is a Gaussian random variable :
¯
cℓj ∼N(E[¯cℓj], σℓjj),
and the positive definite variance covariance matricesVℓ, ℓ=
1,· · ·, k between Gaussian random variables ¯cℓj, j =
1,· · ·, nare given as:
Vℓ=
σℓ11 σℓ12 · · · σℓ1n σℓ21 σℓ22 · · · σℓ2n
· · · · · · · · · · · ·
σℓn1 σℓn2 · · · σℓnn
, i= 1,· · ·, k. (21)
We denote the vectors of the expectation for the random variable row vector¯cℓ as E[¯cℓ] = (E[¯cℓ1],· · ·, E[¯cℓn]), ℓ=
1,· · ·, k.Then, using the variance covariance matrixVℓ, the
objective functionc¯ℓxbecomes a Gaussian random variable.
¯
cℓx∼N(E[c¯ℓ]x,xTVℓx), ℓ= 1,· · ·, k (22)
According to the discussion in the previous section, for a permissible possibility level, MOP4 can be reduced to the following multiobjective stochastic programming problem.
[MOP5(γ)]
min
x∈X (¯c1x+d1(x, γ),· · · ,c¯kx+dk(x, γ)) (23)
where dℓ(x, γ), ℓ = 1,· · ·, k are penalty costs defined by (9). If the decision maker specifies permissible probability levelspℓ, ℓˆ = 1,· · ·, k forc¯ℓx, the multiobjective stochastic
problem (23) can be transformed into the following multiob-jective programming problem through a fractile optimization model [15], [16].
[MOP6(γ,p)ˆ]
min
x∈X(f1(x, γ,pˆ1),· · ·, fk(x, γ,pˆk)) (24)
wherefℓ(x, γ,pˆℓ)is defined as follows.
fℓ(x, γ,pˆℓ)
def
= E[¯cℓ]x+ Φ−1(ˆpℓ)·
√ xTV
ℓx
+dℓ(x, γ) (25)
In MOP6(γ,p)ˆ, it is assumed that Φ−1(·) is an inverse function of a cumulative distribution function for N(0,1) and
0.5 < pℓˆ < 1, ℓ = 1,· · ·, k. It should be noted here that the problem (24) can be regarded as a generalized version of (11), sincefℓ(x, γ,0.5) is equivalent tozℓ(x, γ)ifE[¯cℓ]
is replaced by cℓ, andfℓ(x, γ,pˆℓ), ℓ= 1,· · ·, k are convex
functions with respect tox∈X because of Theorem 3. Similar to Definition 1, we can define a Pareto optimal solution concept to (24).
Definition 2
x∗ ∈ X is said to be a (γ, pˆ)-Pareto optimal solution to MOP6(γ,p)ˆ, if and only if there does not exist anotherx∈
X such that fℓ(x, γ,pℓˆ)≤fℓ(x∗, γ,pℓˆ), ℓ= 1,· · ·, k with strict inequality holding for at least oneℓ.
For the reference objective valuesfℓ, ℓˆ = 1,· · ·, k speci-fied by the decision maker, the corresponding (γ,pˆ)-Pareto optimal solution is obtained by solving the following mini-max problem [13].
[MINMAX2(fˆ,γ,pˆ)]
min
x∈X,λ∈R1λ (26)
s.t. fℓ(x, γ,p)ˆ −fℓˆ ≤λ, ℓ= 1,· · ·, k (27) Similar to Theorem 3, MINMAX2(fˆ, γ, pˆ) becomes a convex programming problem.
The relationships between the optimal solution(x∗, λ∗)of MINMAX2(fˆ, γ,pˆ) and (γ, pˆ)-Pareto optimal solutions to MOP6(γ,p)ˆ can be characterized by the following theorem.
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Theorem 5
(1) If x∗ ∈ X, λ∗ ∈ R1 is a unique optimal solution of MINMAX2(fˆ,γ,pˆ), thenx∗∈X is a (γ,pˆ)-Pareto optimal solution to MOP6(γ,p)ˆ.
(2) If x∗ ∈ X is a (γ, pˆ)-Pareto optimal solution to MOP6(γ,p)ˆ, thenx∗∈X λ∗def= fℓ(x∗, γ,p)ˆ, ℓ= 1,· · ·, k
is an optimal solution of MINMAX2(fˆ, γ, pˆ) for some reference objective valuesfˆ= ( ˆf1,· · ·,fkˆ).
We can present the interactive algorithm to obtain a satisfactory solution from among a (γ, pˆ)-Pareto optimal solution set.
[Algorithm 2]
Step 1: The decision maker sets a permissible possibility level γ and permissible probability levels pℓ,ˆ (0.5 < pℓˆ <
1), ℓ= 1,· · · , k.
Step 2: The decision maker sets the initial reference objec-tive valuesfˆℓ, ℓ= 1,· · · , k forfℓ(x, γ,p)ˆ, ℓ= 1,· · ·, k. Step 3: Solve MINMAX2(fˆ, γ, pˆ) and obtain the corre-sponding optimal solution(x∗, λ∗). For the optimal solution
x∗, a (γ,pˆ)-Pareto optimality test problem is solved.
Step 4: If the decision maker is satisfied with the current value fℓ(x∗, γ,p)ˆ, ℓ = 1,· · ·, k then stop. Otherwise, the
decision maker updates his/her reference objective values
ˆ
fℓ, ℓ = 1,· · · , k, a permissible possibility level γ, and/or permissible probability levelspℓ, ℓˆ = 1,· · ·, k, and return to Step 3.
In order to deal with MOP6(γ,p)ˆ, the decision maker must specify permissible probability levels pˆ in advance. However, in general, the decision maker seems to prefer not only the less value of the objective functionfℓ(x, γ,pˆℓ)
but also the larger value of the permissible probability level
ˆ
pℓ. From such a point of view, we consider the following multiobjective programming problem which can be regarded as a natural extension of MOP6(γ,p)ˆ.
[MOP7(γ)]
min x∈X,pˆℓ∈(0,1),ℓ=1,···,k
(f1(x, γ,pˆ1),· · · , fk(x, γ,pˆk),
−pˆ1,· · ·,−pˆk)
where permissible probability levelspˆℓ, ℓ= 1,· · ·, k are not
fixed values but decision variables.
Considering the imprecise nature of the decision maker’s judgment, we assume that the decision maker has a fuzzy goal for each objective function in MOP7(γ). Such a fuzzy goal can be quantified by eliciting the corresponding mem-bership function. Let us denote a memmem-bership function of an objective function fℓ(x, γ,pˆℓ) as µfℓ(fℓ(x, γ,pˆℓ)), and
a membership function of a permissible probability level pˆℓ
as µpˆℓ(ˆpℓ)respectively. Then, MOP7(γ) can be transformed
into the following problem.
[MOP8(γ)]
max x∈X,pˆℓ∈(0,1),ℓ=1,···,k
(µf1(f1(x, γ,pˆ1)),· · ·, µfk(fk(x, γ,pkˆ )),
µpˆ1(ˆp1),· · ·, µpˆk(ˆpk)) (28)
In the following, we make the following assumptions with respect to the membership functions µfℓ(fℓ(x, γ,pℓˆ)), µpˆℓ(ˆpℓ), ℓ= 1,· · · , k.
Assumption 1
µpˆℓ(ˆpℓ), ℓ= 1,· · ·, k are strictly increasing and continuous
with respect topℓˆ ∈Pℓ def= [ˆpℓmin,pℓˆmax]⊂(0.5,1), where
µpˆℓ(ˆpℓ) = 0 if0≤pℓˆ ≤pℓˆmin, andµpˆℓ(ˆpℓ) = 1 ifpℓˆmax≤ ˆ
pℓ≤1.
Assumption 2
µfℓ(fℓ(x, γ,pℓˆ)), ℓ = 1,· · ·, k are strictly decreasing and continuous with respect to fℓ(x, γ,pℓˆ) ∈ [fℓmin, fℓmax],
where µfℓ(fℓ(x, γ,pℓˆ)) = 0 if fℓ(x, γ,pℓˆ) ≥ fℓmax, and
µfℓ(fℓ(x, γ,pℓˆ)) = 1if fℓ(x, γ,pℓˆ)≤fℓmin.
In order to determine these membership functions appro-priately, let us assume that the decision maker sets pℓˆmin, ˆ
pℓmax,fℓmin andfℓmax as follows.
At first, the decision maker specifies pℓˆmin(> 0.5) and
ˆ
pℓmaxin his/her subjective manner, wherepℓˆminis an
accept-able minimum value andpℓˆmax is a sufficiently satisfactory
minimum value, and sets the intervals :
Pℓdef= [ˆpℓmin,pℓˆmax], ℓ= 1,· · ·, k.
Corresponding to the interval Pℓ, let us denote the inter-val of µfℓ(fℓ(x, γ,pℓˆ)) as [fℓmin, fℓmax], where fℓmin is
a sufficiently satisfactory maximum value and fℓmax is an
acceptable maximum value.fℓmincan be obtained by solving
the following problem.
fℓmin def
= min
x∈Xfℓ(x, γ,pℓˆmin) (29)
In order to obtainfℓmax, we first solve the followingk
pro-gramming problems,minx∈Xfℓ(x, γ,pˆℓmax), ℓ= 1,· · ·, k.
Letxℓ, ℓ= 1,· · ·, k be the corresponding optimal solution.
Using the optimal solutions xℓ, ℓ= 1,· · ·, k,fℓmax can be
obtained as follows.
fℓmax def
= max
ℓ=1,···,k,ℓ̸=ifℓ(xℓ, γ,pℓˆmax) (30)
It should be noted here that, from (25),µfℓ(fℓ(x, γ,pℓˆ))
and µpˆℓ(ˆpℓ) perfectly conflict each other with respect to ˆ
pℓ. Here, let us assume that the decision maker adopts the
fuzzy decision [1], [13], [24] in order to integrate both the membership functionsµfℓ(fℓ(x, γ,pˆℓ))and µpˆℓ(ˆpℓ). Then,
the integrated membership function µDfℓ(x, γ,pˆℓ) can be
defined as follows.
µDfℓ(x, γ,pℓˆ)def= min{µpˆℓ(ˆpℓ), µfℓ(fℓ(x, γ,pℓˆ))} (31)
Using the integrated membership functions µDfℓ(x, γ,pℓˆ),
ℓ= 1,· · ·, k, MOP8(γ) can be transformed into the follow-ing form.
[MOP9(γ)]
max x∈X,pˆℓ∈Pℓ,ℓ=1,···,k
(
µDf1(x, γ,pˆ1),· · ·, µDfk(x, γ,pˆk) )
In order to deal with MOP9(γ), we introduce an M-γ-Pareto optimal solution concept.
Definition 3
x∗ ∈ X,pˆ∗ℓ ∈ Pℓ, ℓ = 1,· · ·, k is said to be an M-γ -Pareto optimal solution to MOP9(γ), if and only if there does not exist another x ∈ X,pℓˆ ∈ Pℓ, ℓ = 1,· · · , k such that µDfℓ(x, γ,pℓˆ) ≥ µDfℓ(x∗, γ,pˆ∗ℓ) ℓ = 1,· · ·, k, with strict inequality holding for at least oneℓ.
For generating a candidate of a satisfactory solution which is also M-γ-Pareto optimal, the decision maker is asked to specify the reference membership values [14]. Once the ref-erence membership values µˆ= (ˆµ1,· · · ,µkˆ )are specified,
IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_14
the corresponding M-γ-Pareto optimal solution is obtained by solving the following minmax problem.
[MINMAX3(µˆ, γ)]
min
x∈X,pˆℓ∈Pℓ,ℓ=1,···,k,λ∈Λ
λ (32)
subject to
ˆ
µℓ−µfℓ(fℓ(x, γ,pˆℓ)) ≤ λ, ℓ= 1,· · · , k (33) ˆ
µℓ−µpˆℓ(ˆpℓ) ≤ λ, ℓ= 1,· · · , k (34)
where
Λdef= [ max
i=1,···,kµiˆ −1,i=1min,···,kµiˆ ].
Because of Assumption 2, the constraints (33) can be trans-formed as follows.
ˆ
pℓ≤Φ
(
µ−f1
ℓ (ˆµℓ−λ)√−E[c¯ℓ]x−dℓ(x, γ) xTV
ℓx
)
(35)
From the constraints (34) and Assumption 1, it holds that
ˆ
pℓ ≥ µ−pˆℓ1(ˆµℓ −λ). Therefore, the constraint (35) can be
reduced to the following inequality where a permissible probability level pˆℓ is disappeared.
µ−f1
ℓ (ˆµℓ−λ)−E[c¯ℓ]x−dℓ(x, γ)
≥ Φ−1(µ−pˆ1
ℓ (ˆµℓ−λ))· √
xTVℓx, ℓ= 1,· · ·, k(36)
Then, MINMAX3(µˆ, γ) can be equivalently reduced to the following problem.
[MINMAX4(µˆ, γ)]
min
x∈X,λ∈Λλ (37)
subject to
µ−f1
ℓ (ˆµℓ−λ)−E[c¯ℓ]x−dℓ(x, γ)
≥ Φ−1(µ−pˆ1
ℓ (ˆµℓ−λ))· √
xTVℓx, ℓ= 1,· · ·, k(38)
In order to solve MINMAX4(µˆ, γ), which is a nonlinear programming problem, we first define the following function for the constraints (38).
gℓ(x, γ, λ)
def = µ−ˆ1
fℓ
(ˆµℓ−λ)−E[c¯ℓ]x−dℓ(x, γ)
−Φ−1(µ−pℓ1(ˆµℓ−λ))· √
xTVℓx, ℓ= 1,· · ·, k (39) It should be noted here that gℓ(x, γ, λ), ℓ = 1,· · · , k are concave functions for any fixed λ∈Λ. Under Assumption 3, it is clear that the constraint set G(λ, γ) is a convex set for anyλ∈[0,1], which is defined as follows.
G(λ, γ)def= {x∈X |gℓ(x, γ, λ)≥0, ℓ= 1,· · ·, k} (40)
The constraint set G(λ, γ) satisfies the following property for λ∈Λ.
Property 1
If λ1, λ2 ∈ Λ and λ1 < λ2, then it holds that G(λ1) ⊂
G(λ2).
(Proof)
From Assumptions 1 and 2, it holds that µ−f1
ℓ (ˆµℓ−λ1) <
µ−f1
ℓ (ˆµℓ−λ2),Φ
−1(µ−1 ˆ
pℓ (ˆµℓ−λ1))>Φ
−1(µ−1 ˆ
pℓ (ˆµℓ−λ2)).
This means thatgℓ(x, γ, λ1)< gℓ(x, γ, λ2)for any x∈X. Therefore,G(λ1, γ)⊂G(λ2, γ)for anyλ1< λ2.
From Property 1, we can easily obtain the optimal so-lution (x∗, λ∗) of MINMAX4(µˆ, γ) by using the bisection
method for λ∈Λ, where it is assumed that G(λmax, γ)̸=
ϕ, G(λmin, γ) =ϕ.
[Algorithm 3]
Step 1: Set λ0=λmax, λ1=λmin andλ←(λ0+λ1)/2.
Step 2: Solve the following convex programming problem forλ∈Λ, and let us denote the optimal solution asx∗∈X.
max
x∈Xgj(x, γ, λ) (41)
subject to
gℓ(x, γ, λ)≥0, ℓ= 1,· · · , k, i̸=j (42)
Step 3: If | λ1−λ0 |< ϵ, then go to Step 4, where ϵ is
a sufficiently small positive constant. If gj(x∗, λ) < 0 or there exists the indexi̸=jsuch thatgℓ(x∗, λ)<0, then set
λ1 ←λ, λ←(λ0+λ1)/2, and go to Step 2. Otherwise, if
gj(x∗, λ)≥0andgℓ(x∗, λ)≥0for anyℓ= 1,· · ·, k, i̸=j, then setλ0←λ, λ←(λ0+λ1)/2, and go to Step 2.
Step 4: Set λ∗ ←λ, and the optimal solution(x∗, λ∗) to MINMAX4(µˆ, γ) is obtained.
The relationship between the optimal solution(x∗, λ∗)of MINMAX4(µˆ, γ)and M-γ-Pareto optimal solutions can be characterized by the following theorem.
Theorem 6
(1) If x∗ ∈ X, λ∗ ∈ Λ is a unique optimal solution to MINMAX4(µˆ, γ), then x∗ ∈ X,pˆ∗ℓ = µ−pˆ1
ℓ (ˆµℓ −λ
∗) ∈ Pℓ, ℓ = 1,· · · , k is an M-γ-Pareto optimal solution to MOP9(γ).
(2) If x∗ ∈ X,pˆℓ∗ ∈ Pℓ, ℓ = 1,· · ·, k is an M-γ -Pareto optimal solution to MOP9(γ), then x∗ ∈ X, λ∗ = ˆ
µℓ−µpˆℓ(ˆp∗ℓ) = ˆµℓ−µfℓ(fℓ(x∗,pˆ∗ℓ)), ℓ = 1,· · · , k is an
optimal solution to MINMAX4(µˆ, γ) for some reference membership valuesµˆ= (ˆµ1,· · ·,µˆk).
Now, following the above discussions, we can present the interactive algorithm in order to derive a satisfactory solution from among an M-γ-Pareto optimal solution set.
[Algorithm 4]
Step 1: The decision maker specifies pℓˆmin > 0.5 and ˆ
pℓmax<1,ℓ= 1,· · · , kin his/her subjective manner. On the
intervalPℓ= [ˆpℓmin,pˆℓmax], the decision maker sets his/her
membership functions µpˆℓ(ˆpℓ), ℓ = 1,· · ·, k according to
Assumption 1.
Step 2: Corresponding to the interval Pℓ, compute fℓmin
and fℓmax by solving the problems (29) and (30). On the
interval[fℓmin, fℓmax], the decision maker sets his/her
mem-bership functionsµfℓ(fℓ(x,pℓˆ)), ℓ= 1,· · ·, k according to Assumption 2.
Step 3: Set the initial reference membership values asµℓˆ = 1, ℓ= 1,· · · , k.
Step 4: Solve MINMAX4(µˆ, γ) to obtain the M-γ-Pareto optimal solution.
Step 5: If the decision maker is satisfied with the current values of the M-γ-Pareto optimal solutionµDfℓ(x∗,pˆ∗ℓ), ℓ=
1,· · ·, k, wherepˆ∗ℓ =µ−pˆ1
ℓ (ˆµℓ−λ
∗), then stop. Otherwise, the
decision maker must update his/her reference membership valuesµℓ, ℓˆ = 1,· · ·, kand/or a permissible possibility level
γ, and return to Step 4.
IV. ACROP PLANNING PROBLEM IN THEPHILIPPINES
In this section, we apply Algorithm 2 to a second crop planning problem of paddy fields in the Philippines [23] under the hypothetical decision maker, in which the water
IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_14
TABLE I
PROFIT COEFFICIENTSctj, t= 1,· · ·,5, j= 1,· · ·,7
j 1 2 3 4 5 6 7
t year ct1 ct2 ct3 ct4 ct5 ct6 ct7 1 1989 4.5 32.6 4.0 72.6 7.3 2.7 10.9 2 1990 5.7 22.7 29.5 13.6 6.3 4.3 24.5 3 1991 3.8 26.3 42.3 42.9 5.1 1.2 13.3 4 1992 3.5 21.3 20.1 35.7 5.2 2.5 26.1 5 1993 4.4 26.2 39.3 22.5 8.4 2.2 26.6
availability constraint in the dry season is expressed as a equality constraint with a fuzzy random variable. In the model farm, only rice (x1) is grown in the wet season (May
to October), and tobacco(x2), tomatoes(x3), garlic(x4), mung
beans(x5), corn(x6) and sweet peppers(x7) are grown in
the dry season (November to April), where xj means the cultivation area (unit: 1 ha) for each cropj= 1,· · ·,7. It is assumed that the farmer has only two persons of available family labor, but he does not have access to hired labor. The farmer must decide the planting ratio among seven kinds of crops (xj, j = 1,· · · ,7) in his/her farmland to maximize
[image:7.595.58.280.67.150.2]his/her total income and minimize total work hours. Table I shows the profit coefficients ctj of seven crops
j (j = 1,· · ·,7) in each year [23]. From Table I, we can compute the expected values as
(E[¯c1], E[¯c2], E[¯c3], E[¯c4], E[¯c5], E[¯c6], E[¯c7]) = (4.38,25.82,27.04,37.46,6.46,2.58,20.28),
and the variance covariance matrix V.
V1=
0.717 0.1005 −0.504 −7.296 0.1005 19.13 −30.13 79.39
−0.504 −30.13 242.06 −239.13
−7.296 79.39 −239.13 515.2 0.4565 2.994 −1.993 −0.217
0.787 −1.250 −5.924 −9.626 0.8745 −26.015 39.27 −143.3
V2=
0.4565 0.787 0.8745 2.994 −1.250 −26.01
−1.993 −5.924 39.27
−0.217 −9.626 −143.3 1.983 0.2665 1.467 0.2665 1.257 3.225 1.467 3.225 57.08
V = (V1|V2)
In the following, we assume that the profit coefficients for seven kinds of crops can be regarded as normal random variables :
¯
cj∼N(E[¯cj], σjj), j= 1,· · · ,7.
Then, the first objective function z¯1(x) (total profit, unit: 1000 pesos) can be defined as follows.
¯
z1(x) def
= 7 ∑
j=1 ¯
cjxj =cx¯ ∼N(E[¯c]x,xTVx)
wherexdef= (x1,· · ·, x7). The second objective function is
total working hours. Table II shows the required working hoursLℓjfor each crop (j = 1,· · ·,7) and each period (from
TABLE II
THE REQUIRED WORKING HOURS FOR EACH PERIODLℓj
period :ℓ Lℓ1 Lℓ2 Lℓ3 Lℓ4 Lℓ5 Lℓ6 Lℓ7 2-May : 1 26
1-Jun : 2 16 2-Jun : 3 160 1-Jul : 4 16 2-Jul : 5 6 2-Aug : 6 8 3-Sep : 7 140
1-Oct : 8 32 8
3-Oct : 9 46 6
1-Nov : 10 36 174 8
2-Nov : 11 100 10 44 12 54 50
3-Nov : 12 22 16 12 8 20
1-Dec : 13 8 38 16 10 16 108
2-Dec : 14 16 94 16 72
3-Dec : 15 8 32 16 24
1-Jan : 16 8 14 64
2-Jan : 17 36 14 12 16
3-Jan : 18 70 6
1-Feb : 19 70 6 180 48
2-Feb : 20 36 14 60 56
3-Feb : 21 36 6 48
1-Mar : 22 36 56
2-Mar : 23 30 32
3-Mar : 24 30
1-Apr : 25 38 56
2-Apr : 26 30
3-Apr : 27 26
the middle ten days in May to the last ten days in April,
ℓ= 1,· · · ,27) [23]. Then, the second objective function (a total number of working hours, unit: 1 hour) can be expressed as :
z2(x) def
= 27 ∑
ℓ=1 7 ∑
j=1
Lℓjxj.
Since the upper limit of the working hours for each period (ℓ= 1,· · ·,27) can be computed as8 (hours)×2(persons)
×10(days)= 160 (hours), the constraints :
7 ∑
j=1
Lℓjxj≤80, ℓ= 1,· · · ,27
must be satisfied. As two land area constraints (unit: 1 ha) for the wet and dry season,
x1≤1, 7 ∑
j=2
xj ≤1, xj≥0, j= 1,· · · ,7
must be satisfied. We assume that the water availability constraint in the dry season is expressed as :
7 ∑
j=2
wjxj =ed,
where the water demand coefficients wj for the crops (j = 2,· · ·,7) are set as (w2, w3, w4, w5, w6, w7) = (264.6,232.3,352.8,88.2,44.1,220.5) [23], and the water supply possible amount is defined as a following LR-type fuzzy random variableed(unit : 1000 gallons).
µe
d(ω)(s) =
L
(
b(ω)−s α
)
, s≤b(ω)
R
(
s−b(ω)
β
)
, s > b(ω)
IAENG International Journal of Applied Mathematics, 46:3, IJAM_46_3_14
where ¯b ∼ N(300,52), α = β = 30, and L(t) = R(t) = 1−t,0≤t≤1.
Since the penalty cost arises only for the shortage of water resource, it is assumed that q+1 = 0, q−1 = 10 and
q+2 =q−2 = 0. Then, for the reference objective valuezℓ, ℓˆ = 1,2 specified by the decision maker, the corresponding γ -Pareto optimal solution is obtained by solving the following minimax problem.
[MINMAX5(fˆ, γ,pˆ)]
max x∈X,λ∈R1λ subject to
f1(x, γ,pˆ)−fˆ1≤λ 27
∑
ℓ=1 7 ∑
j=1
Lℓjxj−fˆ2≤λ
wheref1(x, γ,pˆ)is defined as follows.
f1(x, γ,pˆ) def= − 7 ∑
j=1
E[¯cj]xj+ Φ−01(ˆp)·√xTVx
+d(x, γ)
d(x, γ) def= q1−
(
7 ∑
j=2
wjxj−R−1(γ)β)
·Φ( 7 ∑
j=2
wjxj−R−1(γ)β)
−
∫ ∑7
j=2wjxj−R−1(γ)β
−∞
bϕ(b)db
}
where ϕ(·) andΦ(·) mean the probability density function and the cumulative distribution one forN(300,52), andΦ0(·)
means the cumulative distribution function for N(0,1). Under the hypothetical decision maker, we apply Algo-rithm 2 in section III to the crop planning problem described above. In Step 1, the decision maker sets a permissible possibility level γ = 1 and a permissible probability level
ˆ
p = 0.8. In Step 2, the decision maker sets the initial reference objective values ( ˆf1,fˆ2) = (−33,680). In Step
3, MINMAX5(fˆ,γ,pˆ) is solved by using Mathematica and the corresponding (γ,pˆ)-Pareto optimal solution is obtained. In Step 4, the decision maker is not satisfied with the current value of the (γ, pˆ)-Pareto optimal solution, he/she updates his/her reference objective values as ( ˆf1,fˆ2) = (−33,620),
and return to Step 3. The interactive processes under the hypothetical decision maker is shown in Table III. In this example, a satisfactory solution is obtained at the third iteration, in which a permissible possibility levelγ= 1and a permissible probability level pˆ = 0.8 are fixed at each iteration. For comparison, Table IV shows the interactive processes under the same conditions except that a permissible possibility level is set as γ = 0.5. By comparing Table III for γ= 1with Table IV forγ= 0.5, it is clear that any (γ,
ˆ
p)-Pareto optimal solution forγ= 0.5is superior to (γ,pˆ )-Pareto optimal solution forγ= 1because of the definition of a possibility measure. In any (γ,pˆ)-Pareto optimal solution of Table III and Table IV, only tomatoes (x3) and garlic
(x4) in the dry season and rice (x1) in the wet season are
cultivated. The larger value of a permissible possibility level
γ gives the larger planting ratio of tomatoes (x3) and the
TABLE III
INTERACTIVE PROCESSES FORγ= 1
1 2 3
ˆ
z1 -33 -33 -30
ˆ
z2 680 620 620
ˆ
z1 -33 -33 -30
ˆ
z2 680 620 620
z1(x∗) -27.934 -27.238 -27.204 z2(x∗) 685.07 625.76 622.80 x∗1 0.57343 0.42734 0.42000 x∗2 0.000 0.000 0.000 x∗3 0.55289 0.555327 0.55535 x∗4 0.44465 0.44466 0.44465 x∗5 0.000 0.000 0.000 x∗6 0.000 0.000 0.000 x∗7 0.00246 0.000 0.000
TABLE IV
INTERACTIVE PROCESSES FORγ= 0.5
1 2 3
ˆ
z1 -33 -33 -30
ˆ
z2 680 620 620
z1(x∗) -28.001 -27.305 -27.270 z2(x∗) 685.00 625.70 622.73 x∗1 0.57306 0.42628 0.41894 x∗2 0.000 0.000 0.000 x∗3 0.53228 0.53249 0.53250 x∗4 0.46772 0.46751 0.46750 x∗5 0.000 0.000 0.000 x∗6 0.000 0.000 0.000 x∗7 0.000 0.000 0.000
smaller one of garlic (x4) because of the difference of the
water demand coefficients of tomatoes (x3) and garlic (x4).
V. CONCLUSIONS
In this paper, we formulate three types of multiobjec-tive fuzzy random simple recourse programming problems, in which fuzzy random variables coefficients are involved in equality constraints. In the proposed methods, equality constraints with fuzzy random variables are defined on the basis of a possibility measure and and a two-stage program-ming method. For a given permissible possibility level and reference objective values specified by the decision maker, corresponding minimax problem is solved to obtain a Pareto optimal solution. The proposed method (Algorithm 2) is applied to a farm planning problem in the Philippines, in which an amount supplied of water resource in dry season is represented as a fuzzy random variable and the profit coefficients are represented as random variables. In the near future. we will apply Algorithm 4 to a farm planning problem in the Philippines, in which permissible probability levels are automatically computed through the fuzzy decision.
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![Table I shows the profit coefficients cj (tj of seven cropsj = 1, · · · , 7) in each year [23]](https://thumb-us.123doks.com/thumbv2/123dok_us/381816.535641/7.595.58.280.67.150/table-i-shows-prot-coefcients-seven-cropsj-year.webp)
